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A033577 a(n) = (3*n+1) * (4*n+1). 8

%I

%S 1,20,63,130,221,336,475,638,825,1036,1271,1530,1813,2120,2451,2806,

%T 3185,3588,4015,4466,4941,5440,5963,6510,7081,7676,8295,8938,9605,

%U 10296,11011,11750,12513,13300,14111,14946,15805,16688,17595,18526,19481,20460,21463

%N a(n) = (3*n+1) * (4*n+1).

%C Also the 120ยบ spoke (or ray) of a hexagonal spiral of Ulam. - _Robert G. Wilson v_, Jul 06 2014

%C If two independent real random variables x and y are distributed according to the same exponential distribution with pdf(x) = lambda * exp(-lambda * x) for some lambda > 0, then the probability that 3 <= x/(n*y) < 4 is given by n/a(n) for n>1. - _Andres Cicuttin_, Dec 11 2016

%H Colin Barker, <a href="/A033577/b033577.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Colin Barker_, Dec 12 2016: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.

%F G.f.: (1 + 17*x + 6*x^2) / (1 - x)^3.

%F (End)

%e See A056105 example section for hexagonal spiral of Ulam diagram. - _Robert G. Wilson v_, Jul 06 2014

%p A033577:=n->(3*n+1)*(4*n+1): seq(A033577(n), n=0..50); # _Wesley Ivan Hurt_, Jul 06 2014

%t f[n_] := (3n + 1)(4n + 1); Array[f, 50, 0] (* _Robert G. Wilson v_, Jul 06 2014 *)

%o (PARI) vector(50, m, 12*m^2 - 17*m + 6) \\ _Michel Marcus_, Jul 06 2014

%o (PARI) Vec((1 + 17*x + 6*x^2) / (1 - x)^3 + O(x^50)) \\ _Colin Barker_, Dec 12 2016

%o (MAGMA) [(3*n+1)*(4*n+1) : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 06 2014

%Y Subsequence of A281333.

%Y Cf. A056105, A244802, A056106, A244803, A056107, A244804, A056108, A244805, A056109, A244806, A003215.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Wesley Ivan Hurt_, Jul 06 2014

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Last modified December 15 03:08 EST 2017. Contains 296020 sequences.